Having used polymake to research this topic, I provided the recommended polymake citations at the end. Also, I made use of homogenous coordinates, a la polymake, which simplified the presentation of the answer.

Each polytope defined in the Q&A is essentially a certain simplex placed on top of a hypercube. I show that these polytopes are self-dual by showing a facet-vertex incidence matrix for each which is symmetric. Interestingly, a Sierpinski triangle pattern forms in the NW quadrant of the incidence matrix.

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